Triangle Congruence Theorems Geometry Triangle Congruence Theorems

Congruent Triangles Congruent triangles have three congruent sides and and three congruent angles. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles.

The Triangle Congruence Postulates &Theorems AAS ASA SAS SSS FOR ALL TRIANGLES LA HA LL HL FOR RIGHT TRIANGLES ONLY

Theorem If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent. Think about it… they have to add up to 180°.

A closer look... If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. 85° 30° But do the two triangles have to be congruent?

Example Why aren’t these triangles congruent? 30° 30° Why aren’t these triangles congruent? What do we call these triangles?

So, how do we prove that two triangles really are congruent?

the 2 triangles are CONGRUENT! ASA (Angle, Side, Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . . F E D A C B then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! AAS (Angle, Angle, Side) Special case of ASA If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . . F E D A C B then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! SAS (Side, Angle, Side) If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . . F E D A C B then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! SSS (Side, Side, Side) F E D A C B In two triangles, if 3 sides of one are congruent to three sides of the other, . . . then the 2 triangles are CONGRUENT!

Warning: No SSA Postulate There is no such thing as an SSA postulate! Side Angle Side The triangles are NOTcongruent!

There is no such thing as an SSA postulate! Warning: No SSA Postulate There is no such thing as an SSA postulate! NOT CONGRUENT!

If we know that the two triangles are right triangles! BUT: SSA DOES work in one situation! If we know that the two triangles are right triangles! Side Side Side Angle

These triangles ARE CONGRUENT by HL! We call this HL, for “Hypotenuse – Leg” Remember! The triangles must be RIGHT! Hypotenuse Hypotenuse Leg RIGHT Triangles! These triangles ARE CONGRUENT by HL!

the 2 triangles are CONGRUENT! HL (Hypotenuse, Leg) If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . A C B F E D then the 2 triangles are CONGRUENT!

There is no such thing as an AAA postulate! Warning: No AAA Postulate There is no such thing as an AAA postulate! Different Sizes! Same Shapes! E B A C F D NOT CONGRUENT!

Congruence Postulates and Theorems SSS SAS ASA AAS AAA? SSA? HL

Example 1 A C B Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D E F

Example 2 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B F E D

Example 3 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D A C B

Example 4 SAS Why are the two triangles congruent? B C D E F Why are the two triangles congruent? What are the corresponding vertices? SAS A   D C   E B   F

Example 5 SSS Why are the two triangles congruent? What are the corresponding vertices? SSS D B A   C ADB   CDB C ABD   CBD

Example 6 B C D A Given: Are the triangles congruent? Why? S

Example 7 R H S Given: Are the Triangles Congruent? Why? Q R S P T Given: mQSR = mPRS = 90° Are the Triangles Congruent? Why? R H S QSR  PRS = 90°

Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS – Pairs of congruent angles and the side not contained between them. SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides

Summary --- for Right Triangles Only: HL – Pair of sides including the Hypotenuse and one Leg

THE END!!!